(1) Field of the Invention
This invention relates to a computer-aided method for signal or data processing and more particularly to a method for finding, by means of nonlinear regression analyses, a Probability Density Function (PDF) for arbitrary exponential functions falling into one of four classes, the underlying probability density function and data structures conforming to the exponential model.
(2) Description of the Prior Art
Of the many continuous probability distributions encountered in signal processing, a good number are distinguished by the fact that they are derived from exponential functions on the time interval of 0 to ∞, e.g., failure rate distributions, Poisson processes, Chi-Square, gamma, Rayleigh, Weibull, Maxwell and others involving exponential functions. Such an exponential function is also used in O""Brien et al. (U.S. Pat. No. 5,537,368) to generate a corrected data stream from the raw data stream of a sensor.
Occasionally modeling involves functions for which the probability density function (PDF) and its moments need to be derived de novo. Often times, research scientists and engineers are confronted with modeling a random variable x when the probability density function (PDF) is unknown. It may be known that the variable can be reasonably well approximated by a gamma density. Then solving a problem under the assumption that x has a gamma density will provide some insight into the true situation. This approach is all the more reasonable since many probability distributions are related to the gamma function. However, deriving the PDF and its statistical moments using the standard approach involving moment generating functions (MGF) and complex-variable characteristic functions is difficult and somewhat impractical to implement in applied research settings. The complexity of current methods for constructing the PDF and MGF limits the class of models used for analyzing correlated data structures. O""Brien et al. (U.S. Pat. No. 5,784,297) provided a method for finding a probability density function (PDF) and its statistical moments for an arbitrary exponential function of the form g(x)=xcex1xmexe2x88x92xcex2xn, 0 less than x less than ∞, where xcex1, xcex2, n greater than 0, m greater than xe2x88x921 are real constants in one-dimensional distributions and g(x1,x2, . . . ,x1) in the hyperplane. However, the method in the ""297 patent is based on a single probability model within the domain 0xe2x86x92∞, thus limiting its application.
Accordingly, it is a general purpose and object of the present invention to provide a computer-aided method for determining density and moment functions for a useful class of exponential functions in signal processing.
Another object of the present invention is to provide a method for constructing the PDF and MGF which offers the possibility of constructing the PDF and MGF for a larger class of such functions.
A still further object is to enhance standard assumptions about the structure of error or disturbance terms by including a larger class of models to choose from.
These objects are provided with the present invention by a simple substitution method for finding a probability density function (PDF) and its statistical moments for a chosen one of four newly derived probability models for an arbitrary exponential function of the forms g(x)=xcex1xmexe2x88x92xcex2xn, xe2x88x92∞ less than x less than ∞;                     g        ⁡                  (          x          )                    =              α        ⁢                  xe2x80x83                ⁢                  x          m                ⁢                  ⅇ                      -                          βx              n                                            ,                  0        ≤        x         less than         ∞            ;                          g        ⁢                  (          x          )                    =                                    α            ⁡                          (                                                x                  -                  a                                b                            )                                m                ⁢                  ⅇ                      -                                          β                ⁡                                  (                                                            x                      -                      a                                        b                                    )                                            n                                            ,                            -          ∞                 less than         x         less than         ∞            ;              xe2x80x83            ⁢      and                          g        ⁡                  (          x          )                    =                                    α            ⁡                          (                                                x                  -                  a                                b                            )                                m                ⁢                  ⅇ                      -                                          β                ⁡                                  (                                                            x                      -                      a                                        b                                    )                                            n                                            ,          xe2x80x83        ⁢          0      ≤      x       less than               ∞        .            
The model chosen will depend on the domain of the data and whether information on the parameters a and b exists. These parameters may typically be the mean or average of the data and the standard deviation, respectively. For example, it may be known that the signal of interest within the data being processed has a domain from xe2x88x92∞xe2x86x92∞ and a typical mean and standard deviation. Thus a model of the third form would be used.
Once the model is chosen, computer implemented non-linear regression analyses are performed on the data distribution to determine the solution set Sn(xcex1n,mn,xcex2n,n) beginning with n=1. A root-mean-square (RMS) is calculated and recorded for each order of n until the regression analyses produce associated RMS values that are not changing in value appreciably. The basis function is reconstructed from the estimates in the final solution set and a PDF for the basis function is obtained utilizing methods well known in the art. The MGF, which characterizes any statistical moment of the distribution, is obtained using a novel function derived by the inventors and the mean and variance are obtained in standard fashion. Once the parameters xcex1, xcex2, m and n have been determined for a set of data measurements through the system identification modeling, the PDF-based mean and variance are determinable, and simple binary hypotheses may be tested.
By the inclusion of four newly derived models, the method of the present invention provides a choice of models from a larger and more useful class of exponential functions covering the full domain (xe2x88x92∞xe2x86x92∞). The method of the present invention further provides enhanced standard assumptions about the structure of error or disturbance terms by the use of additional variables such as mean and standard deviation parameters.